Transcript Classifications with Application in Enterprise Credit

Polynomial Radial Basis Function: A
New Local Similarity Measure
Linkai Luo, Ting Liu, Hong Peng , Qifeng Zhou ,
Xiamen University, China. 361005
Tel: (86)-592-2580181
e-mail: [email protected]
Outline
 Introduction
 Polynomial Radial Basis Function(PRBF)
 Comparisons between GRBF and PRBF

Conclusions and future work
Introduction
• A good similarity measure is crucial in pattern
recognition problem
• Local similarity measure
• Current research of local similarity measure
• Analysis of Gaussian Radial Basis Function
(GRBF)
Local similarity measure
• Local similarity measure means similarity measure is
meaning only in local neighborhood, similarity
measure is unmeaning (designed to zero ) out of local
neighborhood.
• Local similarity measure can be designed more easily
– We can construct the similarity measure more easily if data
points are near. If data points are far apart, we often don’t
know what similarity measure is suitable for them.
• Local similarity measure can reduce the
computational cost.
Current research of local similarity
measure
• The main researches are how to define suitable
similarity measure by combining with specific
domain knowledge [11-14].
• The general local similarity measure is needed when
specific domain knowledge is lacking.
• There are only a few literatures to investigate general
local similarity measure.
– Luca Cazzanti and Maya R. Gupta [15] proposed a local
similarity discriminant analysis (local SDA) to solve
classification problem, but the computational cost didn’t be
considered in their work.
Gaussian Radial Basis Function
(GRBF)
 || x  x ' || 
g ( x, x ')  exp 

2



2
Analysis of GRBF
• Gaussian Radial Basis Function (GRBF)
g(x,x’)=exp{-σ-2||x-x’||2} is a famous similarity
measure function which has been widely used in
supervised leaning, unsupervised learning, and semisupervised learning
• GRBF essentially is a global similarity measure
function although the function vale of GRBF tends to
zero when data points are far infinitely. When data
points are far away, the quantity of their similarity
measure often is unuseful, and it will waste the
computation time.
Analysis of GRBF
• Hence constructing a similarity
measure function, which is of similar
properties as GRBF inside some
neighborhood and equals to zero
outside some neighborhood, is
worthy and significative.
The proposed properties of new local
similarity measure function
• Local GRBF
– It is similar as GRBF inside some neighborhood ,
and equals to zero outside this neighborhood.
• A faster computational rapidity than GRBF.
• At least one-order differentiable.
PRBF
• From polynomial function set, we obtain a
polynomial function which satisfies above
properties by solving a interpolation problem.

 4 (|| x  x ' ||2  2 )2， | x | 
p( x, x ')  
| x | 

0，
• We name it as Polynomial Radial Basis
Function(PRBF)
The picture of PRBF
Fig.1 The pictures of GRBF and PRBF
Comparisons between GRBF and PRBF
Tab.1 The comparisons of some properties between Gaussian RBF and
Polynomial RBF
Property
Gaussian RBF
Polynomial RBF
Local similarity property
No
Yes
Computational rapidity
Slow
Fast
Differentiable property
Infinitely
differentiable
One-order
differentiable
The comparison of computational rapidity
between GRBF and PRBF
• In PRBF, only three times of multiplication and
division, as well as one time of addition are needed
when the function value is computed at a time.
• In GRBF, the times of multiplication and division
depend on the value of independent variable. The
times of multiplication and division will increase if
the value of independent variable increases. Generally
the times of multiplication and division are more than
three.
• PRBF is faster than GRBF.
Fig.2 The comparison of computational rapidity
between GRBF and PRBF
The performance comparison of similarity measure to
learning machine between GRBF and PRBF
• Consider a semi-supervised learning problem.
• Compare PRBF with GRBF on three benchmark data
set “Heart”, “Breast” and “Australian”.
• Label Propagation algorithm is selected to solve
semi-supervised learning problem.
• Select respectively one third and rest two third of
samples as labeled samples and unlabeled samples for
all three data set.
• The comparison results are listed in Tab.2.
Tab.2 Comparisons of testing accuracy between GRBF and PRBF for semisupervised leaning problem
Data Set
Testing accuracy
of GRBF
Testing accuracy
of PRBF
Heart
81.85
82.01
Breast
93.31
94.45
Australian 85.37
85.56
 Remake:The testing accuracy is the average testing accuracy on ten
group random test for all three data set.
 The testing accuracy of PRBF are as same as
that of GRBF, even it precede GRBF a little.
Conclusions
• PRBF is locally similar to GRBF
• PRBF proposes a faster computational rapidity than
GRBF.
• PRBF proposes almost same similarity measure
ability as GRBF.
• In a word, PRBF is a good substitute for
GRBF if high computational rapidity is
demanded.
Future work
• Further application tests for PRBF (such as
clustering, classification, etc.).
• The research for local similarity measure with
positive semi-definite property.
Thank you！